$\Sigma$-Subsets of Natural Numbers

It is shown that the class of all possible families of $\Sigma$-subsets of finite ordinals in admissible sets coincides with a class of all non-empty families closed under $e$-reducibility and $\oplus$. The construction presented has the property of being minimal under effective definability. Also, we describe the smallest (w.r.t. inclusion) classes of families of subsets of natural numbers, computable in hereditarily finite superstructures. A new series of examples is constructed in which admissible sets lack in universal $\Sigma$-function. Furthermore, we show that some principles of classical computability theory (such as the existence of an infinite non-trivial enumerable subset, existence of an infinite computable subset, reduction principle, uniformization principle) are always satisfied for the classes of all $\Sigma$-subsets of finite ordinals in admissible sets